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It shall be a fault if in play, the shuttle touches any object (...) outside the court.

In singles, the post is outside of the court, so it would be a fault.In doubles, the post could be argued as being inside the court, as it should be right on the sidelines, and that seems to be the intention of rule 1.5:

The posts or its supports shall not extend into the court beyond the side lines

On the other hand, rule 15.1 explicitly says

strikes the (...) post and starts to fall towards the surface of the court on the striker's side

I would infer from the latter rule that in doubles, it counts just as if the shuttle would have rolled over the net if the part of the post that's hit is on the sideline.In many (amateur) settings, the net posts are actually outside of the court. In that case, it's a fault.

The post is considered as part of the net.
So if the shuttle lands on top of the post and falls down the opponents side (passes to the other side of the net) and lands within the side lines (regardsless singles/doubles) the point is won by the the one who made the stroke.
So yes the same applies as if it had just hit the net.

Well it is not mentioned directly in the rules anywhere (that I could find), but I remember this from my days as an EBU umpire.
This was actually one of the first things we learned in the first umpire cource I ever took

Regarding rule 15.1, then you need to read the hole rule and not just a part of it.
Rule 15.1 states:

15. SHUTTLE NOT IN PLAYA shuttle is not in play when:

15.1 it strikes the net or post and starts to fall towards the surface of the court on the striker’s side of the net;

This rule only applies when the shuttle hits the net/post and do not cross over to the other side, which is not the case in BernieR's question.

Yes, I'm fully aware that rule 15.1 is unrelated, but why does it distinguish between net and post if the post is always part of the net? Good to know that that this is handled informally in practice, but that seems like the kind of thing one should include in the rules .

1.4 The posts shall be 1.55 metres in height from the surface of the court and shall remain vertical when
the net is strained as provided in Law 1.10

1.5 The posts shall be placed on the doubles side lines as in Diagram A irrespective of whether singles or
doubles is being played. The posts or its supports shall not extend into the court beyond the side lines.

If you look at these 2 equipment rules you will find that the net posts are supposed to be exactly the same height as the net and of course the shuttle has to pass over the net. There is no need to distinguish in this case between net and post because if it struck the top of the post it is also hitting the top of the net.

In law 15.1 it is necessary to mention because usually the shuttle is "not in play" until it hits the ground, but not so in 15.1. If you hit the net post further down it has nothing to do with the net and it needed to be made clear in this instance.

If it had hit the net cord and fallen into the opponents' court, the player who struck the shuttle would have won the point. So does the same apply if it hits the top of the support post?

It is implied in the inquiry that this is during a rally.

Be that as it may, it is not automatic that the player who struck that shuttle will always win that rally. S/he could have committed an infraction, like touching the net, or worse, throwing a racquet, or distracted opponent, or other unsporting conduct.

The other part needs clarification, specifically, the top of the support post. Is this clearly above the net? We use volleyball posts in the gym to tie the net, and these supports are clearly above the required height of 5'1". Then, if shuttle strikes that support which is above the net-level and then falls inside the opponent court, fault.

In tournament conditions, the net supports are part of the net and the uprights are lined with the outside boundary of the tramlines. These uprights will be 5'1" at their highest point, are are part of that court. One could essentially hit the shuttle from outside (hitter's side of the court), hit this post and have that shuttle land in the opponent's court. Good rally.

The other part needs clarification, specifically, the top of the support post. Is this clearly above the net? We use volleyball posts in the gym to tie the net, and these supports are clearly above the required height of 5'1". Then, if shuttle strikes that support which is above the net-level and then falls inside the opponent court, fault.

In tournament conditions, the net supports are part of the net and the uprights are lined with the outside boundary of the tramlines. These uprights will be 5'1" at their highest point, are are part of that court. One could essentially hit the shuttle from outside (hitter's side of the court), hit this post and have that shuttle land in the opponent's court. Good rally.

If you using support posts that are non standard, then these rules don't apply. But then again, I doubt you use these volleyball posts in an actual tournament. Thus, for games like these, use some common sense and a consensus. Or even play a let.

As for the second point, yes. Absolutely right. It would be a good rally.

In singles, the post is outside of the court, so it would be a fault.In doubles, the post could be argued as being inside the court,

I agree with it.

Just to share, I recalled last year while representing my company in a mixed doubles match where the last point of the 1st game. I send the shuttle to hit the top of the pole and the umpire counted that in and awarded the match to me. However he failed to realise something else. My opponent raise the alarm, so do I. Why? because the shuttle landed outside the court and then roll inside the court. Umpire didn't see it at all and insisted play goes on start of 2nd match insisting that shuttle hitting the pole is legal but fail to realise it went outside the court before going in.

So I brought this question up to a BWF referee who has done the Olympics in the past.

The response I got was, in singles, if it were to hit the pole, it would still be a good shot. She states that she infers the "object outside the court" to be such as lighting or maybe basketball hoops for example that is outside the court.

The only likely way for this to happen is possibly a smash that hits the inside corner of the pole and bouncing in. It is highly unlikely for the shuttle to hit the pole, roll up and over the pole and 1 1/2 ft into the singles court.